Introduction / Context:
This problem checks fluency with decimal multiplication and powers of 10. Multiplying several decimals often leads to misplaced decimal points. A structured approach using scientific notation prevents such mistakes and ensures an exact final answer.
Given Data / Assumptions:
- Expression: 3 × 0.3 × 0.03 × 0.003 × 30
- All quantities are exact decimals (no rounding required).
- Standard arithmetic precedence applies (only multiplication here).
Concept / Approach:
Convert decimals to powers of 10 whenever helpful. Group convenient factors to simplify mental arithmetic. Track exponents carefully, because each decimal place corresponds to a negative power of 10 (for example, 0.3 = 3 × 10^-1).
Step-by-Step Solution:
Group integers: 3 × 30 = 90Convert decimals: 0.3 × 0.03 × 0.003 = (3×10^-1) × (3×10^-2) × (3×10^-3)Multiply digits: 3 × 3 × 3 = 27Add exponents: 10^(-1-2-3) = 10^-6So 0.3 × 0.03 × 0.003 = 27 × 10^-6 = 0.000027Now multiply by 90: 90 × 0.000027 = 0.00243
Verification / Alternative check:
Count decimal places directly: there are 1 + 2 + 3 = 6 decimal places in the three small factors; 3 × 30 = 90 scales the product by 90, giving the same 0.00243.
Why Other Options Are Wrong:
- .0000243 and .000243: Under-count decimal shift.
- .0243: Over-shifts the decimal by two places.
Common Pitfalls:
- Losing track of exponents when multiplying several decimals.
- Rounding mid-way instead of keeping exact values until the end.
Final Answer:
0.00243
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